2. Benchmark solution

2.1. Calculation method used for the reference solution

2.1.1. Modeling A

The test focuses on the value of the error indicator at the output of RAFF_XFEM. Note \(I(M)\) the value of this indicator at any point \(M\).

At point \({P}_{1}\) at the bottom left of the structure, the indicator is the opposite of the distance to the nearest notch point, which is \(I({P}_{1})\mathrm{=}\mathrm{-}({\mathit{CP}}_{1}\mathrm{-}{R}_{e})\).

At point \({P}_{2}\) at the bottom right of the structure, the indicator is the opposite of the distance to the nearest notch point, which is \(I({P}_{2})\mathrm{=}\mathrm{-}({\mathit{DP}}_{2}\mathrm{-}{R}_{e})\).

At point \({P}_{3}\) at the top right of the structure, the indicator is the opposite of the distance to the right end of the rightmost crack, i.e. \(I({P}_{3})\mathrm{=}\mathrm{-}A\text{'}{P}_{3}\) or \(A\text{'}\mathrm{=}A+\frac{{L}_{A}}{2}\overrightarrow{x}\).

At point \({P}_{4}\) at the top left of the structure, the indicator is the opposite of the distance to the left end of the leftmost crack, i.e. \(I({P}_{4})\mathrm{=}\mathrm{-}B\text{'}{P}_{4}\) or \(B\text{'}\mathrm{=}B\mathrm{-}\frac{{L}_{B}}{2}\overrightarrow{x}\).

2.1.2. B and C models

The test focuses on the value of the diameter of the smallest mesh. If \({h}_{0}\) is the initial mesh size, \({h}_{c}\) is the target mesh size after refinement, then the minimum number of calls to Lobster to reach \({h}_{c}\) is \(\text{nb\_raff}\mathrm{=}E(n)+1\), with \(n\mathrm{=}\frac{\mathrm{ln}({h}_{0})\mathrm{-}\mathrm{ln}({h}_{c})}{\mathrm{ln}(2)}\). After refinement, the most refined mesh size is \(h\mathrm{=}\frac{{h}_{0}}{{2}^{\text{nb\_raff}}}\).

2.2. Benchmark results

2.2.1. Modeling A

With the numerical values used in the test, we find:

\(\begin{array}{c}I({P}_{1})\mathrm{=}\mathrm{-}(\sqrt{{\mathrm{0,25}}^{2}+{\mathrm{0,2}}^{2}}\mathrm{-}\mathrm{0,05})\mathrm{\approx }\mathrm{-}\mathrm{0,27015621187164246}\\ I({P}_{2})\mathrm{=}\mathrm{-}(\sqrt{{\mathrm{0,25}}^{2}+{\mathrm{0,2}}^{2}}\mathrm{-}\mathrm{0,05})\mathrm{\approx }\mathrm{-}\mathrm{0,27015621187164246}\\ I({P}_{3})\mathrm{=}\mathrm{-}\sqrt{{\mathrm{0,25}}^{2}+{\mathrm{0,1}}^{2}}\mathrm{\approx }\mathrm{-}\mathrm{0,26925824035672524}\\ I({P}_{4})\mathrm{=}\mathrm{-}\sqrt{{\mathrm{0,25}}^{2}+{\mathrm{0,2}}^{2}}\mathrm{\approx }\mathrm{-}\mathrm{0,32015621187164245}\end{array}\)

2.2.2. B and C models

With \({h}_{0}\mathrm{=}\frac{\sqrt{(2)}}{20}\) and \({h}_{c}\mathrm{=}\frac{{h}_{0}}{10}\), we get \(h\mathrm{=}\mathrm{0,0044194}\).